SECT. 4] SOUND SCATTERING BY MARINE ORGANISMS 523 



equation (38) may be re-written 



^ ' Ec ^ ' d\ogz ^ ' 



Early in the 1950's reverberation from deep scattering layers was found to 

 be strongly frequency-dependent (Hersey, Johnson and Davis, 1952). Ac- 

 cordingly the total scattered energy must be analyzed within bands of fre- 

 quencies small compared with the scale of the frequency dependence, and yet 

 broad enough and so designed that their response time, which controls the 

 effective pulse width t, preserves T<^t. 



The total energy, Et, oi the shock wave was computed by Machlup from the 

 empirical formulae given by Arons (1954) 



P = 2.16(104)(lf/3/P)i-i3p.s.i. 



= 58(10-6)lf'/3(lf'/3/i2)-o.22sec, 



where P is the peak pressure, ® is the decay time of the shock wave, which is 

 approximately exponential, W is the weight of TNT in pounds, and R is the 

 mean range in feet of a region for which E is sought. It is 



Et = 4:77 \ F{t) dr - 477 i?2 — ^-^r/e ^^ 

 J pc Jo 



pc 2 

 The filter transfer characteristic was taken to be 



Y{w) = bl[b + iico-wo)], 



where co is angular frequency, ojq the "center" angular frequency of the filter, 

 and the half-power (3-dB-down) points are cuo ± b. That is, the bandwidth of 

 the filter is 6/77 cycles per second. The total energy, E, of equation (38) may be 

 considered the fraction of the total energy that would be passed by the filter. 

 This fraction, A/^Et, is computed as the square of the output voltage of the 

 filter for an input having the same time dependence as the shock wave. Mach- 

 lup's approximation (Machlup and Hersey, 1955) is 



where a = b. 



1 \2 



tan 8 = {-^ — b\ OJQ, coo = Stt/q. 



Then E of equation (38) may be written 



E = A/^Et' 



